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5 CONDITION OF CONTINUITY OF A MULTILINEAR MAPPING 105

(5.5.4) Let Ebea normed space, F a Banach space, G a dense subspace o/E,
a continuous linear mapping of G into F. Then there is a unique continuous
linear mapping J of E into F which is an extension off.

From (5.5.1) it follows that / is uniformly continuous in G, since
\\f(x)-f(y)\\ = \\f(x-y)\\^a-\\x-y\\; hence by (3.15.6) there is a
unique continuous extension / of / to E. The fact that / is linear follows
from (5.1.5) and the principle of extension of identities (3.15.2).

PROBLEMS

1. Let u be a mapping of a normed space E into a normed space F such that u(x + y) —
u(x) -f */0>)forany pair of points*, y of E and that uis bounded in the ball B(0; l)inE;
show that u is linear and continuous. (Observe that u(rx) = ru(x) for rational r, and

that for ^eB(0; I),M( *+->>) -«(*)< -||«(y)|| for every integer n^ 1; conclude

II \ ⁿ / II ⁿ

that u(Xx) — \u(x) for every real A, by taking y = n(r — A)JC, where r is rational.)

2. Let E, F be two normed spaces, u a linear mapping of E into F. Show that if for every
sequence (x_n) in E such that lim x_n — 0, the sequence («(*„)) is bounded in F, then u

II-+00

is continuous. (Give an indirect proof.)

3. (a) Let a, b be two points of a normed space E. Let BI be the set of all x e E such that
||x —a|| = ||jt — b\\ = \\a — b\\/2i for n> 1, let B_n be the set of xEB_n-i such that
II* -y\\< 8(&_n-i)l2 for all y e B_B_i (8(A) being the diameter of a set A). Show that
8(B_B) =s$ 8(B_n_i)/2, and that the intersection of all the B_n is reduced to (a + 6)/2.

(b) Deduce from (a) that if/is an isometry of a real normed space E onto a real normed
space F, then f(x) = u(x) + c, where u is a linear isometry, and c e F.

4. Let us call rectangle in N x N a product of two intervals of N; for any finite subset H of
NX N, let i/r(H) be the smallest number of rectangles whose union is H. Let (H_B) be an
increasing sequence of finite subsets of N x N, whose union is N x N and such that the
sequence (</<**„)) is bounded. Let E, F, G be three normed spaces, (*„) (resp. (/„)) a
convergent series in E (resp. F),/a continuous bilinear mapping of E x F into G.Show
that

(*) lim 2 f(xn,yk) =

5. Let (H_B) be an increasing sequence of finite subsets of N x N, whose union is N x N; for
each ; e N and each n e N, let yd, n) be the smallest number of intervals of N whose union
is the set H_B-^l (j) of all integers i such that (/, 7) e H_B. Suppose <p(/, n) is bounded in N x N.
Let (x_n) be a convergent series in a normed space E, 0>_B) an absolutely convergent series
in a normed space F, u a continuous bilinear mapping of E x F into a normed space G.
Show that formula (*) of Problem 4 still holds (use (5.5.1), and remark that the sums

2 xi are bounded in E for all;, n) (cf. Section 12.16, Problem 12).

6. Let E,"F be two real normed spaces. A mapping/of E into F is said to be linear in a
neighborhood 0/0 if there exists a S > 0 such that: (1) the relations ||x|| ^ 8, ||*'U < 8,
HJC 4- x'll < 8 in E imply f(x + x') =/(*) +/(*'); (2) the relations ||*|| ^ 8, ||A*|| < 8
in E (with A 6 R) imply /(A*) = A/(x).



	5 CONDITION OF CONTINUITY OF A MULTILINEAR MAPPING 105 (5.5.4) Let Ebea normed space, F a Banach space, G a dense subspace o/E, a continuous linear mapping of G into F. Then there is a unique continuous linear mapping J of E into F which is an extension off. From (5.5.1) it follows that / is uniformly continuous in G, since \\f(x)-f(y)\\ = \\f(x-y)\\^a-\\x-y\\; hence by (3.15.6) there is a unique continuous extension / of / to E. The fact that / is linear follows from (5.1.5) and the principle of extension of identities (3.15.2).

	PROBLEMS 1. Let u be a mapping of a normed space E into a normed space F such that u(x + y) — u(x) -f /0>)forany pair of points, y of E and that uis bounded in the ball B(0; l)inE; show that u is linear and continuous. (Observe that u(rx) = ru(x) for rational r, and that for ^eB(0; I),M( +->>) -«()< -\|\|«(y)\|\| for every integer n^ 1; conclude II \ ⁿ / II ⁿ that u(Xx) — \u(x) for every real A, by taking y = n(r — A)JC, where r is rational.) 2. Let E, F be two normed spaces, u a linear mapping of E into F. Show that if for every sequence (x_n) in E such that lim x_n — 0, the sequence («(„)) is bounded in F, then u II-+00* is continuous. (Give an indirect proof.) 3. (a) Let a, b be two points of a normed space E. Let BI be the set of all x e E such that \|\|x —a\|\| = \|\|jt — b\\ = \\a — b\\/2i for n> 1, let B_n be the set of xEB_n-i such that II* -y\\< 8(&_n-i)l2 for all y e B_B_i (8(A) being the diameter of a set A). Show that 8(B_B) =s$ 8(B_n_i)/2, and that the intersection of all the B_n is reduced to (a + 6)/2. (b) Deduce from (a) that if/is an isometry of a real normed space E onto a real normed space F, then f(x) = u(x) + c, where u is a linear isometry, and c e F. 4. Let us call rectangle in N x N a product of two intervals of N; for any finite subset H of NX N, let i/r(H) be the smallest number of rectangles whose union is H. Let (H_B) be an increasing sequence of finite subsets of N x N, whose union is N x N and such that the sequence (</<*„)) is bounded. Let E, F, G be three normed spaces, („) (resp. (/„)) a convergent series in E (resp. F),/a continuous bilinear mapping of E x F into G.Show that () lim 2 f(xn,yk) =* 5. Let (H_B) be an increasing sequence of finite subsets of N x N, whose union is N x N; for each ; e N and each n e N, let yd, n) be the smallest number of intervals of N whose union is the set H_B-^l (j) of all integers i such that (/, 7) e H_B. Suppose <p(/, n) is bounded in N x N. Let (x_n) be a convergent series in a normed space E, 0>_B) an absolutely convergent series in a normed space F, u a continuous bilinear mapping of E x F into a normed space G. Show that formula () of Problem 4 still holds (use (5.5.1), and remark that the sums 2 xi are bounded in E for all;, n) (cf. Section 12.16, Problem 12). 6. Let E,"F be two real normed spaces. A mapping/of E into F is said to be linear in a neighborhood 0/0 if there exists a S >* 0 such that: (1) the relations \|\|x\|\| ^ 8, \|\|'U < 8, HJC 4- x'll < 8 in E imply f(x* + x') =/() +/('); (2) the relations \|\|\|\| ^ 8, \|\|A\|\| < 8 in E (with A 6 R) imply /(A*) = A/(x).